Spectral measures of small index principal graphs

Details Spectral measures of small index principal graphs Spectral measures of small index principal graphs

2006-03-24

arxiv.org/abs/math/0603572v3

Comm. Math. Phys. 269 (2007), 259-281

Mathematics

Operator Algebras

23 pages

Scientific paper

10.1007/s00220-006-0122-1

The principal graph $X$ of a subfactor with finite Jones index is one of the important algebraic invariants of the subfactor. If $\Delta$ is the adjacency matrix of $X$ we consider the equation $\Delta=U+U^{-1}$. When $X$ has square norm $\leq 4$ the spectral measure of $U$ can be averaged by using the map $u\to u^{-1}$, and we get a probability measure $\epsilon$ on the unit circle which does not depend on $U$. We find explicit formulae for this measure $\epsilon$ for the principal graphs of subfactors with index $\le 4$, the (extended) Coxeter-Dynkin graphs of type $A$, $D$ and $E$. The moment generating function of $\epsilon$ is closely related to Jones' $\Theta$-series.

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